AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
and 
Show that
Where 
and 
Show that
Where
, where 
and 
are (not necessarily distinct) positive integers. The integer 
is written as the sum of 
Tiranian integers. What is the smallest possible value of ?
be non-nagative real numbers such that 
.
.
and its internal angle bisector 
. The line intersects the circumcircle 
of triangle at 
and 
. Circle 
with diameter 
cuts again at 
. Prove that 
is the symmedian line of triangle .
be a triangle with circumcircle 
. Arbitrary points 
on 
respectively. Circumcircle 
of triangle 
intersects at 
. 
intersects 
at 
. 
cuts and at 
respectively. Construct parallelogram 
. Prove that 
are collinear.
be an isosceles triangle with 
, and let be its incenter. Incircle touches sides 
at 
, respectively. Foot of altitudes from 
to 
are 
, respectively. Rays 
intersect 
at 
, respectively. Prove that 
touches incircle at 
.
with two altitudes and .Let be the point on the line such that 
.The lines 
and 
intersect at point 
, and 
is the point on segment 
such that 
.Suppose that bisects .Prove that triangle 
is isosceles.
-plane given points 
on the planes 
respectively. Let 
be the intersection point of the line 
and the 
-plane.
. When the point 
moves on the perimeter of the circle with center 
, radius 1 on the plane 
,.
and 
on the plane 
. When the point 
moves on the perimeter of the circle with center 
, radius 1 on the plane and the point moves on the perimeter of the square 
, draw the domain swept by the point on the -plane, then find the area.
with 
, let its incircle be tangent to sides , 
, and 
at , 
, and 
, respectively. The internal angle bisector of 
intersects lines 
and 
at and , respectively. Let 
and 
be distinct points on side such that 
. Finally, let 
be the circumcircle of 
. is tangent to the circumcircle of 
. is tangent to the incircle of . with their incenter . Point lies on on the same side as 
wrt 
. Point lies on the shorter arc of the circumcircle . It is given that 
Prove that 
is the angle bisector of 
.
of the triangle touch its side at . Let the line intersect at 
and denote the excentre of opposite to 
by 
. Let 
and 
be the midpoints of and 
respectively.
and 
are concyclic.
be a triangle with 
. is the midpoint of 
. Let the angle bisector of 
cut 
at and 
be the centroid of . 
meets at 
and is the projection of onto . Prove that 
lie on a common circle.
the unit circle, so 
. Suppose line 
cuts at points 
and 
.
, and similarly 
.
.
, so, by triangle inequality, we have 
, and we're done.
.
